Inhomogeneous minima of mixed signature lattices

The euclidean minimum of a number field is a real number, which measureshow far the
number field is from being euclidean with respect to the norm. A still open conjecture of
Minkowski predicts an upper bound for the euclidean minima of totally real number
fields in terms of the degree and the discriminant.
I will explain how a topological method due to McMullen can be used to produce upper
bounds for euclidean minima of arbitrary number fields in terms of signature and discriminant.